The conductivity of subterranean formations or strata which surround a well (borehole) varies depending on such factors as the porosity of the formation and the amount and character of any fluids which might permeate or be trapped in the formation. The magnitude of the formation conductivity is used to deduce certain characteristics of the strate penetrated by the well and to assist in predicting the likely performance in yield of the well. This information is useful with respect to a wide variety of wells, including oil and gas wells, geothermal wells, groundwater wells, mining wells and others. Some useful information is provided by a knowledge of the relative differences in conductivity, i.e., a knowledge that a given stratum is more or less conductive than another stratum or knowledge regarding the ratio of the conductivities of two layers or formations. The most useful information, however, is the absolute, rather than relative, value of conductivity of a formation, expressible in units such as mhos per meter.
Because of the usefulness of this information, several methods have been devised in an attempt to obtain absolute values of conductivity (resistivity), many employing measurements of electrical parameters made in the well itself. These methods have had some success in providing conductivity information in uncased wells or in uncased portions of partially cased wells. However, these methods cannot be used to find absolute values of conductivity of strata surrounding cased portions of wells, because such methods were designed to function properly and accurately only in uncased wells and such methods do not work in cased wells.
Methods for determining formation conductivity in the vicinity of cased wells would be useful in a number of commonly-encountered situations including exploration of old oil fields, mapping injection profiles in the secondary recovery of oil and gas, and special conditions of formations when it is necessary to use full or partial casing directly after drilling.
Although a number of attempts have been made to apply electric logging techniques to cased wells, only nuclear and acoustic methods are in widespread use in cased wells. Proposals for electrical methods for use in cased wells include U.S. Pat. No. 2,459,196 to Stewart entitled "Electrical Logging Method and Apparatus", and U.S.S.R. Pat. No. 56,026 to Alpin entitled "Method of the Electrical Logging in Wells with Casing".
The Stewart reference discloses a method for deducing relative change of resistivities of strata penetrated by a cased borehole by measuring current flowing along various intervals of the casing using current electrodes for supplying current to the casing, and which are electrically connected to a current source, and receiver electrodes which are positioned along the path of current through the casing. In principle, from these measurements one can obtain information about the formation conductivity. However, the approach described in this patent does not allow one to solve this problem practically for the following reasons: (a) the method does not recognize that there is a range of optimal relative distances between the current electrodes and the receiver electrodes and between the receiver electrodes or that variations in casing thickness or conductivity and deviation of the distance between receiver electrodes from the ideal distance produce significant effects on the measured quantities; (b) the magnitude of current within every interval of the casing is measured separately; correspondingly, every measurement is performed with different accuracy due to several factors such as a change in amplifier gain, a change of grounding resistance with the casing, and the instability of the current source; and (c) inasmuch as the leakage of current into the formation is very small with respect to a measured current, it is virtually impossible in such a way to evaluate this leakage with proper accuracy, derived as a difference of measured currents particularly when the formation thickness is relatively small. The effect of each of these considerations is that the value of current "leakage" (i.e. that current which leaks through the casing and into the formation, rather than traveling along the casing), is comparable to or less than the magnitude of "noise" signals.
The Alpin reference discloses connecting a current source directly to the casing at one position and measuring the voltage drop across probes directly contacting the casing at other positions. The Alpin method uses two current electrodes for supplying current to the casing and which are electrically connected to the current source. Alpin also provides receiver electrodes for use in receiving or measuring the magnitude of electrical parameters in the borehole. One current electrode is positioned relatively close to the receiver electrodes and the other current electrode is positioned relatively far removed from the receiver electrodes. By repeating the voltage measurement at a number of levels, Alpin obtains a curve which is intended to be applied for the same purposes as the usual curve of apparent resistivity. The approach described by Alpin is impractical for providing useful information about formation conductivity because: (a) this approach does not recognize that variations in casing thickness and conductivity and deviation of the distance between receiver electrodes from the ideal distance produces significant effects on measured quantities; (b) only relative conductivities are theoretically obtainable; and (c) Alpin does not recognize that a minimum distance is required between the current source electrodes and receiver electrodes to eliminate the influence of distortions of the electrical field on the voltage measurements.
The method and apparatus disclosed in copending parent application Ser. No. 740,734 has certain similarities to the method and apparatus described herein, however, the method of the parent application does not recognize that variations in casing thickness and casing conductivity and deviation of the distance between receiver electrodes from the ideal distance produces significant effects on measured quantities, and therefore does not disclose a method for overcoming or compensating for these effects.
In general, previous methods for electrical logging in cased wells can in principle produce data but such data suffers from the characteristic that the signal to noise ratio is so low that useful information regarding the surrounding formations cannot be reliably obtained. An understanding of the methods which result in data with improved signal to noise ratio requires some discussion of the characteristics of an electrical field residing in a cased borehole and the surrounding formation.
Understanding of the electrical field configuration is assisted by considering a cased well at least several hundred meters long having a current electrode situated approximately on the longitudinal axis of the well and located several hundred feet below the surface. When current is supplied to the current electrode, current close to the location of the current electrode, i.e., within a distance less than a few times the radius of the borehole, flows radially, i.e., symmetrically in all directions. At distances from the current electrode of about ten to twenty times greater than the radius of the borehole, the current is practically oriented parallel to the axis of the borehole. Although some of the current still flows through the borehole medium, the significant majority of the current is conducted by the well casing at this distance from the current electrode. The electric field along the borehole axis can be approximately defined by the following equation: ##EQU1## where:
E.sub.z (L) is the electric field at a distance L from the current electrode;
I is the current at the current electrode;
S.sub.c is the casing conductance;
S.sub.o is the borehole conductance;
a is the borehole radius; and
.sigma..sub.f is the formation conductivity.
The relative magnitude of the three terms of this relation depends heavily on the value of (L/a). In the region relatively close to the current electrodes, when the value of (L/a) is less than about ten, the electric field is mainly defined by the second term of Equation (1). At greater distances from the current electrodes, when (L/a) is greater than about ten, but while the current flow is essentially parallel to the borehole axis, the field is dominated by the first and last terms of Equation (1).
If receiving electrodes for measuring voltage are inserted into the borehole and positioned at points in the "intermediate" range (i.e., in the area where current flow is substantially parallel to the borehole axis) based on a number of theoretical considerations, the voltage between receiver electrodes M and N will be given with a high accuracy by Equation (2). ##EQU2## where:
V.sub.MN is the voltage measured between receiver electrodes M and N, where N is situated above M;
MN is the distance between electrodes M and N;
S.sub.MN is the casing conductance between M and N;
.alpha.=(.pi..S.sub.MN).sup.-1/2 ;
.rho. is the formation resistivity; and
L.sub.MN is the distance from the current electrode to the midpoint of the interval MN.
If a third voltage measuring or receiver electrode M.sub.1 is positioned above electrode N, which in turn is in a position above electrode M, Equation (2) can be used to derive an expression for the formation resistivity: ##EQU3## where: ##EQU4##
It has been found that Equation (3) establishes a relation between resistivity of a medium and the value of .DELTA.V.sub.le, provided that the following assumptions are made: (1) the distance between electrodes M and N is equal to the distance between electrodes N and M.sub.1 ; (2) the casing resistivity between electrodes M and N is equal to the casing resistivity between electrodes N and M.sub.1 ; and (3) the voltage measurements are made in the borehole where the current flow is substantially parallel to the borehole axis.
As follows from the theory of the method, if the above three assumptions are met, then the ratio of the formation resistivity laterally adjacent to a first voltage measurement location to the formation resistivity laterally adjacent to a second voltage measurement location can be expressed as: ##EQU5## where
.rho..sub.2 is the resistivity of the formation laterally adjacent to location 2;
.rho..sub.1 is the resistivity of the formation laterally adjacent to location 1;
.DELTA.V.sub.le.sup.(1) =V.sub.MN -V.sub.NM.sbsb.1 when electrodes M, N and M.sub.1 are in a first fixed relationship relative to each other (position 1); and
.DELTA.V.sub.le.sup.(2) =V.sub.MN -V.sub.NM.sbsb.1 when electrodes M, N and M.sub.1 are in a second fixed relationship relative to each other (position 2).
Whether it is justified to simplify Equation (3) so as to be able to express the ratio of formation conductivities in terms of measurable voltages, as in Equation (4), depends on the validity of the three assumptions above. It is possible to estimate the magnitude of the effect which violations of assumptions (1) and (2) would have on measured voltages. Variations in the distance between the electrodes can be caused by such factors as thermal expansion or contraction and high pressures developed in the borehole. Differences in the casing conductivity can be caused by such factors as a change in the casing thickness or a change in the resitivity of the casing material or the presence of fractures, any of which can be caused by manufacturing variances or corrosion taking place in the borehole. When the magnitude of the effect of such variations on voltage measurements is compared to the magnitude of the voltage signal which it is desired to measure, it is found that voltages which arise from violation of the above assumptions are of the same magnitude as the voltage which it is desired to measure. In other words, in ordinary borehole conditions, violation of assumptions (1) and (2) produces "noise" which is substantially equal in magnitude to the signal which is being measured. Thus, one of the difficulties to be solved in order to provide reliable indications of formation resistivity is compensation for variation in electrode spacing and/or casing conductivity.